How to read between the lines of electronic spectra: the diagnostics of fluctuations in strongly correlated electron systems

Thomas Schäfer,Alessandro Toschi

doi:10.1088/1361-648x/abeb44

Abstract

While calculations and measurements of single-particle spectral properties often offer the most direct route to study correlated electron systems, the underlying physics may remain quite elusive, if information at higher particle levels is not explicitly included. Here, we present a comprehensive overview of the different approaches which have been recently developed and applied to identify the dominant two-particle scattering processes controlling the shape of the one-particle spectral functions and, in some cases, of the physical response of the system. In particular, we will discuss the underlying general idea, the common threads and the specific peculiarities of all the proposed approaches. While all of them rely on a selective analysis of the Schwinger–Dyson (or the Bethe–Salpeter) equation, the methodological differences originate from the specific two-particle vertex functions to be computed and decomposed. Finally, we illustrate the potential strength of these methodologies by means of their applications the two-dimensional Hubbard model, and we provide an outlook over the future perspective and developments of this route for understanding the physics of correlated electrons.

Highlights

Electronic correlations at the one- and two-particle levelThe major challenge to be faced when studying quantum systems with a high degree of correlations between their constituents is the difficulty of disentangling the information of a single particle from the rest of the system
Though sharing similar goals and -to some extentphilosophy, these novel approaches can be grouped in two main classes, depending on whether they are based (i) on a direct decomposition of the full vertex functions within the parquet formalism (“parquet decomposition” [9], heuristically associated to the motto “divide et impera”, or rather (ii) on changes of representation of the Schwinger-Dyson equation for the self-energy (“fluctuation diagnostics” [8], under the motto “mutatis mutandis”)
From a more technical perspective, we have discussed how the applicability of the first diagnostic tool gets typically restricted to the perturbative regime, because of the almost ubiquitous occurrence of divergences of 2PI vertices in many-electron problems [23, 28, 24, 9, 26, 19, 20, 27]

Summary

Introduction

The major challenge to be faced when studying quantum systems with a high degree of correlations between their constituents is the difficulty of disentangling the information of a single particle from the rest of the system. We observe that since Λ2PI represents the cornerstone of the parquet equation, its divergences are expected to considerably impact all procedures based on a parquet decomposition This is exactly what happens: By increasing U into the intermediate-coupling regime, as it is the case for the DMFT calculations for the Hubbard model on a cubic lattice shown in the left panel of Fig. 5, it becomes very hard to extract significant information from the parquet decomposition, as the parquet-decomposed contributions to Σ start displaying [9] wild oscillations in all the scattering channels (ch, pp, as well -- Λ) where divergences of 2PI vertices have been encountered§. Before discussing the impact of such restrictions of the parquet decomposition at the end of the section, we will illustrate here literature as well as new applications of this post-processing technique in the “safe” weakto-intermediate regime, aiming at highlighting the usefulness and versatility of this approach in its region of full applicability

Versatility of the parquet decomposition

The challenge of the non-perturbative regime

Versatility of the fluctuation diagnostics

Conclusions and outlook